attention, coordination, and bounded recallapa522/slides...is endogenous precision and r s = p s h s...
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Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Attention, Coordination, and Bounded Recall
Alessandro Pavan
Northwestern University
Chicago FED, February 2016
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Motivation
Many socioeconomic environments
- large group of agents
- actions under dispersed information
Useful modelization for:
- production or network externalities
- incomplete markets
- business cycles
- large Cournot-Bertrand games
- elections
...
Most of the literature: exogenous information structure
Many phenomena of interest: attention (info. acquisition) is central
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Motivation
Many socioeconomic environments
- large group of agents
- actions under dispersed information
Useful modelization for:
- production or network externalities
- incomplete markets
- business cycles
- large Cournot-Bertrand games
- elections
...
Most of the literature: exogenous information structure
Many phenomena of interest: attention (info. acquisition) is central
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Motivation
Many socioeconomic environments
- large group of agents
- actions under dispersed information
Useful modelization for:
- production or network externalities
- incomplete markets
- business cycles
- large Cournot-Bertrand games
- elections
...
Most of the literature: exogenous information structure
Many phenomena of interest: attention (info. acquisition) is central
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Motivation
Many socioeconomic environments
- large group of agents
- actions under dispersed information
Useful modelization for:
- production or network externalities
- incomplete markets
- business cycles
- large Cournot-Bertrand games
- elections
...
Most of the literature: exogenous information structure
Many phenomena of interest: attention (info. acquisition) is central
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
This paper
Flexible (yet rich) framework
- complementarity or substitutability in actions
- rich set of payoff interdependencies
Equilibrium and effi cient allocation of attention
- perfect recall
- bounded recall
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
This paper
Flexible (yet rich) framework
- complementarity or substitutability in actions
- rich set of payoff interdependencies
Equilibrium and effi cient allocation of attention
- perfect recall
- bounded recall
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Questions
What payoff interdependencies create ineffi ciency in eq. allocation ofattention?
How does ineffi ciency in attention relate to ineffi ciency in use ofinformation?
What is the effect of bounded recall?
What policies can alleviate such ineffi ciencies? (related work)
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Related literature (incomplete)
Effi cient use of information and social value of informationRadner (1977), Vives (JET 1984, 2013)
Morris and Shin (AER 2002)
Angeletos and Pavan (AER, 2004, Ecma 2007, Jeea, 2009)
...Information acquisition/(in)attention in coordination settingsVives and Van Zandt (2007)
Hellwig and Veldkamp (Restud, 2009)
Amir and Lazzati (2014)
Mackowiak and Wiederholt (AER, 2009, 2012)
→ Myatt and Wallace (Restud 2012)
Szkup and Trevino (2013), Yang (2013)
→ Colombo, Femminis and Pavan (Restud 2014)
Tirole (2014), Denti (2016)
...MemoryBenabou Tirole (JPE 2004)
Wilson (2004), Kocer (2010)
...Analogy-based equilibriumJehiel (JET 2005)
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Plan
1 Model (perfect recall)
2 Equilibrium allocation of attention
3 Effi cient allocation of attention
4 Bounded recall
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Plan
1 Model (perfect recall)
2 Equilibrium allocation of attention
3 Effi cient allocation of attention
4 Bounded recall
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Plan
1 Model (perfect recall)
2 Equilibrium allocation of attention
3 Effi cient allocation of attention
4 Bounded recall
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Plan
1 Model (perfect recall)
2 Equilibrium allocation of attention
3 Effi cient allocation of attention
4 Bounded recall
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
ModelActions and gross payoffs
ui
(ki , {k j} j 6=i, θ
)
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
ModelActions and gross payoffs
Continuum of agents with payoffs:
u
(k , K , θ , σ
2k
)where:k ∈ R — individual action
K =∫
k′dΨ(k′) —aggregate action
σ2k =
∫(k′−K)2dΨ(k′) —dispersion
θ ∈ R —underlying uncertainty ("fundamentals")
Assumptions:u(·) quadratic in (k,K,θ), linear in σ2
k
u(·) s.t. equilibrium and first-best unique and bounded
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Examples
Investment spillovers (Angeletos and Pavan AER 2004)
ui = Rki− c(ki)
R= (1−a)θ +aK and c(ki) =1
2k2
i
Beauty contest (Morris and Shin AER 2002)
ui =−(1− r) · (ki−θ)2− r · (L(ki)− L)
L(ki)≡∫ (
k′− ki
)2dΨ(k′)= (ki−K)2+σ
2k and L=
∫L(k)dΨ(k)= 2σ
2
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Examples
Monetary economies (Woodford 2005, Colombo, Femminis and Pavan,2014, Llosa and Venkateswaran, 2015)
u(θ ,Ci,Ni)≡V (Ci)−Ni
Ci =
(∫[0,1]
cv−1
v
hidh
) vv−1
Yi = θα Ni
∫[0,1]
phchidh≤ piYi−T
Cournot and Bertrand games (Vives JET 1984)
ui = (a−θK) · ki−1
2k2
i
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
ModelInformation and attention
Common prior:θ ∼ N(0,π−1
θ)
N = 1,234,576 sources of information:
yl = θ + ε l with ε l ∼ N(0,η−1l) l = 1, ...,N
Agent i’s "impressions" xi = (xil)Nl=1 with
xil = yl +ξ
il with ξ
il ∼ N
(0,(
zil · tl)−1
)l = 1, ...,N
where
η l : accuracy
tl : transparency/clarity
zil : attention
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
ModelInformation and attention
Common prior:θ ∼ N(0,π−1
θ)
N = 1,234,576 sources of information:
yl = θ + ε l with ε l ∼ N(0,η−1l) l = 1, ...,N
Agent i’s "impressions" xi = (xil)Nl=1 with
xil = yl +ξ
il with ξ
il ∼ N
(0,(
zil · tl)−1
)l = 1, ...,N
where
η l : accuracy
tl : transparency/clarity
zil : attention
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
ModelInformation and attention
Common prior:θ ∼ N(0,π−1
θ)
N = 1,234,576 sources of information:
yl = θ + ε l with ε l ∼ N(0,η−1l) l = 1, ...,N
Agent i’s "impressions" xi = (xil)Nl=1 with
xil = yl +ξ
il with ξ
il ∼ N
(0,(
zil · tl)−1
)l = 1, ...,N
where
η l : accuracy
tl : transparency/clarity
zil : attention
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
ModelAttention cost and net payoffs
Attention cost: C(zi) where zi = (zil)Nl=1
· C′n(zi)> 0, all zi 6= 0
· limzn→∞C′n(zi) = ∞
· convex (results extend to concave, e.g., entropy reduction)
E.g. C(zi) = c(∑l zi
l
)E.g. C(zi) = ∑l g(zi
l)
...but also C(zi) = µ(zi;y) (entropy reduction)
Net payoff
u
(ki,K,σ
2k ,θ)−C(zi)
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
ModelTiming
agents allocate attention zi
update their beliefs based on xi
commit their actions ki
payoffs realized
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Plan
1 Model (perfect recall)
2 Equilibrium allocation of attention
3 Effi cient allocation of attention
4 Bounded Recall
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Equilibrium use of information (Angeletos and Pavan, Ecma 2007)
Optimality:k j = E[ κ+α(K−κ) | x j ; z j]
where
κ = κ0+κ1θ (complete-info. equilibrium action)
α ≡ ukK
|ukk | −→ equilibrium degree of coordination
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Equilibrium allocation of attention
Theorem
There exists a unique symmetric equilibrium. In this eq., the attention z thateach agent assigns to the various sources of information is s.t., for any sourcen= 1, ...,N that receives strictly positive attention,
zn = κ1γn
√|ukk|
2C′n(z)tn
where
γn ≡(1−α)πn
1−αρn
πθ +∑Ns=1
(1−α)πs
1−αρs
is "influence" of the source
and where
πs =ηszsts
zsts+ηs
is endogenous precision and ρs =πs
ηs
is endogenous "publicity"
Given equilibrium allocation of attention z, equilibrium actions are given by
ki = κ0+κ1
(∑
Nn=1 γnxi
n
)all i ∈ [0,1], almost all xi ∈ RN .
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Private value of attention
Envelope reasoning: hold k(·; z) fixed
Agent’s eq. continuation payoff (fixing k(·; z)):
Ui(zi; z) = E[u(K,K,σ k,θ)]+
ukk
2Var[ki−K | zi, z,k(·; z)]−C(zi)
Private value of attention
−|ukk|2· ∂Var[k−K | z,k(·;z)]
∂ zn
private aversion to dispersion · reduction in dispersion(fixing eq. strategy k(·;z))
Result generalizes Colombo, Femminis, Pavan (Restud 2014)
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Private value of attention
Envelope reasoning: hold k(·; z) fixed
Agent’s eq. continuation payoff (fixing k(·; z)):
Ui(zi; z) = E[u(K,K,σ k,θ)]+
ukk
2Var[ki−K | zi, z,k(·; z)]−C(zi)
Private value of attention
−|ukk|2· ∂Var[k−K | z,k(·;z)]
∂ zn
private aversion to dispersion · reduction in dispersion(fixing eq. strategy k(·;z))
Result generalizes Colombo, Femminis, Pavan (Restud 2014)
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Private value of attention
Envelope reasoning: hold k(·; z) fixed
Agent’s eq. continuation payoff (fixing k(·; z)):
Ui(zi; z) = E[u(K,K,σ k,θ)]+
ukk
2Var[ki−K | zi, z,k(·; z)]−C(zi)
Private value of attention
−|ukk|2· ∂Var[k−K | z,k(·;z)]
∂ zn
private aversion to dispersion · reduction in dispersion(fixing eq. strategy k(·;z))
Result generalizes Colombo, Femminis, Pavan (Restud 2014)
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Private value of attention
Envelope reasoning: hold k(·; z) fixed
Agent’s eq. continuation payoff (fixing k(·; z)):
Ui(zi; z) = E[u(K,K,σ k,θ)]+
ukk
2Var[ki−K | zi, z,k(·; z)]−C(zi)
Private value of attention
−|ukk|2· ∂Var[k−K | z,k(·;z)]
∂ zn
private aversion to dispersion · reduction in dispersion(fixing eq. strategy k(·;z))
Result generalizes Colombo, Femminis, Pavan (Restud 2014)
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Plan
1 Model (perfect recall)
2 Equilibrium allocation of attention
3 Effi cient allocation of attention
4 Bounded Recall
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi ciency
Welfare : ex-ante utility of representative agent
Definition
Effi cient allocation consists of attention z∗ along with action rule k∗(·;z∗) thatjointly maximize
E[u(k,K,σ2k ,θ) | z]−C(z)
Team problem
Planner’s problem: control incentives but cannot transfer information
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi cient use of information (Angeletos and Pavan, Ecma 2007)
Given attention z, effi ciency in actions requires that k∗(·;z) solves
k∗(x;z) = E [κ∗+α∗(K−κ
∗) | x ; z ] ∀x,
whereκ∗ = κ
∗0+κ
∗1θ −→ FB
α∗ ≡ uσσ −2ukK −uKK
ukk+uσσ
= 1− aversion to volatilityaversion to dispersion
socially optimal degree of coordination
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi cient allocation of attention
Theorem
Effi ciency in attention requires that, for any n for which z∗n > 0,
z∗n = κ∗1γ∗n
√|ukk+uσσ |2C′n(z∗)tn
where
γ∗n ≡
(1−α∗)πn
1−αρn
πθ +∑Ns=1
(1−α∗)πs
1−α∗ρs
is effi cient "influence" of the source
πs =ηsz∗s ts
z∗s ts+ηs
is endogenous precision and ρs =π∗sηs
is endogenous publicity
Recall that eq.
zn = κ1γn
√|ukk|
2C′n(z)tn
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi cient allocation of attention
Theorem
Effi ciency in attention requires that, for any n for which z∗n > 0,
z∗n = κ∗1γ∗n
√|ukk+uσσ |2C′n(z∗)tn
where
γ∗n ≡
(1−α∗)πn
1−αρn
πθ +∑Ns=1
(1−α∗)πs
1−α∗ρs
is effi cient "influence" of the source
πs =ηsz∗s ts
z∗s ts+ηs
is endogenous precision and ρs =π∗sηs
is endogenous publicity
Recall that eq.
zn = κ1γn
√|ukk|
2C′n(z)tn
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi cient allocation of attention
Envelope reasoning
Welfare under effi cient use of information (for given attention z)
w∗(z)≡ E[u(κ∗,κ∗,0,θ)]−L ∗(z)−C(z),
where u(κ∗,κ∗,0,θ) is welfare under FB allocation and
L ∗(πx,πz)≡|ukk+2ukK +uKK |
2Var[K−κ
∗ | k∗(·;z),z]
+|ukk+uσσ |
2Var[k−K | k∗(·;z),z]
Holding k∗(·;z), Var[K−κ∗ | k∗(·;z),z] independent of z
Social value of attention
−|ukk+uσσ |2
· ∂Var[k−K | z,k∗(·;z)]∂ zn
social aversion to dispersion · reduction in dispersion(fixing eff. strategy k∗(·;z) )
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi cient allocation of attention
Envelope reasoning
Welfare under effi cient use of information (for given attention z)
w∗(z)≡ E[u(κ∗,κ∗,0,θ)]−L ∗(z)−C(z),
where u(κ∗,κ∗,0,θ) is welfare under FB allocation and
L ∗(πx,πz)≡|ukk+2ukK +uKK |
2Var[K−κ
∗ | k∗(·;z),z]
+|ukk+uσσ |
2Var[k−K | k∗(·;z),z]
Holding k∗(·;z), Var[K−κ∗ | k∗(·;z),z] independent of z
Social value of attention
−|ukk+uσσ |2
· ∂Var[k−K | z,k∗(·;z)]∂ zn
social aversion to dispersion · reduction in dispersion(fixing eff. strategy k∗(·;z) )
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi cient allocation of attention
Envelope reasoning
Welfare under effi cient use of information (for given attention z)
w∗(z)≡ E[u(κ∗,κ∗,0,θ)]−L ∗(z)−C(z),
where u(κ∗,κ∗,0,θ) is welfare under FB allocation and
L ∗(πx,πz)≡|ukk+2ukK +uKK |
2Var[K−κ
∗ | k∗(·;z),z]
+|ukk+uσσ |
2Var[k−K | k∗(·;z),z]
Holding k∗(·;z), Var[K−κ∗ | k∗(·;z),z] independent of z
Social value of attention
−|ukk+uσσ |2
· ∂Var[k−K | z,k∗(·;z)]∂ zn
social aversion to dispersion · reduction in dispersion(fixing eff. strategy k∗(·;z) )
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi cient allocation of attention
Envelope reasoning
Welfare under effi cient use of information (for given attention z)
w∗(z)≡ E[u(κ∗,κ∗,0,θ)]−L ∗(z)−C(z),
where u(κ∗,κ∗,0,θ) is welfare under FB allocation and
L ∗(πx,πz)≡|ukk+2ukK +uKK |
2Var[K−κ
∗ | k∗(·;z),z]
+|ukk+uσσ |
2Var[k−K | k∗(·;z),z]
Holding k∗(·;z), Var[K−κ∗ | k∗(·;z),z] independent of z
Social value of attention
−|ukk+uσσ |2
· ∂Var[k−K | z,k∗(·;z)]∂ zn
social aversion to dispersion · reduction in dispersion(fixing eff. strategy k∗(·;z) )
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Equilibrium vs effi cient allocation of attention
Private value of attention
−|ukk|2· ∂Var[k−K | z,k(·;z)]
∂ zn
private aversion to dispersion · reduction in dispersion(fixing eq. strategy k(·;z))
Social value of attention
−|ukk+uσσ |2
· ∂Var[k−K | z,k∗(·;z)]∂ zn
social aversion to dispersion · reduction in dispersion(fixing eff. strategy k∗(·;z) )
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Equilibrium vs effi cient allocation of attention
Private value of attention
−|ukk|2· ∂Var[k−K | z,k(·;z)]
∂ zn
private aversion to dispersion · reduction in dispersion(fixing eq. strategy k(·;z))
Social value of attention
−|ukk+uσσ |2
· ∂Var[k−K | z,k∗(·;z)]∂ zn
social aversion to dispersion · reduction in dispersion(fixing eff. strategy k∗(·;z) )
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi cient allocation of attention
Effi ciency in attention requires
- effi ciency in use of information: k(·;z) = k∗(·;z)
- private = social aversion to dispersion ⇔ uσσ = 0
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Plan
1 Model (perfect recall)
2 Equilibrium allocation of attention
3 Effi cient allocation of attention
4 Bounded Recall
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Bounded Recall
Idea: posteriors correct, but agents cannot recall influence of individualsources
Given attention z j, posterior beliefs about θ continues to be Normal withmean
x j =∑N
n=1δ nxi
n, with δ n ≡πn
πθ +∑Ns=1 πs
and πs ≡ηszsts
zsts+ηs
and precision πθ +∑Ns=1 πs
However, agent is unable to decompose x j into various impressionsx j ≡ (x j
1, ...,xjN).
Equivalently, unable to decompose his posteriors into
θ | xin
Measurability constraint on k(x j;zz)
Distinction relevant only in strategic setting
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Bounded Recall
For simplicity: πθ = 0
Theorem
In unique symmetric equilibrium, given allocation z#, actions given by
ki = κ0+κ1xi
For any source that receives strictly positive attention in eq.,
C′n(z#) =−|ukk|
2
∂Var[k−K;z#,k(·;z#)
]∂ zn
− |ukk|2(1−α)
∂Var[K−κ;z#,k(·;z#)
]∂ zn
Novel effect:
−|ukk|2(1−α)
∂Var[K−κ;z#,k(·;z#)
]∂ zn
private aversion to volatility of own’s average action · reduction in volatility(fixing eq. strategy k#(·;z))
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Bounded vs Perfect Recall
Theorem
Let z be eq. allocation of attention with perfect recall. There exist publicitythresholds ρ ′,ρ ′′ ∈ [0,1] s.t., starting from z, any agent with bounded recall isbetter off by
(a) locally increasing attention to sources for which ρn ∈ [ρ ′,ρ ′′];
(b) locally decreasing attention to sources for which ρn /∈ [ρ ′,ρ ′′] .
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Bounded vs Perfect Recall
Reallocation of attention towards sources of average (endogenous)publicity
ρn =zsts
zsts+ηs
Sources of low publicity: useful to forecast θ
Sources of high publicity: useful to forecast K
Sources of intermediate transparency: good compromises
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Bounded vs Perfect Recall
Previous result about best responses extends to equilibrium
Suppose C(z) = c(∑
Ns=1 zs
)Theorem
Let z be eq. attention with perfect recall and z# eq. attention with boundedrecall. There exist thresholds t ′, t ′′ ∈ R++ s.t. z#
n > zn only if tn ∈ [t ′, t ′′] .Furthermore for any n for which tn ∈ [t ′, t ′′], z#
n < zn only if z#n = 0.
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Effi ciency under Bounded Recall
...see paper!
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Conclusions
Attention in large economies with
- complementarity / substitutability in actions
- rich set of payoff interdependencies
- rich information structure
Effi ciency in allocation of attention requires
(a) absence of externalities from action-dispersion
(b) effi ciency in use of information
Bounded recall: reallocation of attention towards sources withintermediate transparency
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Conclusions
Attention in large economies with
- complementarity / substitutability in actions
- rich set of payoff interdependencies
- rich information structure
Effi ciency in allocation of attention requires
(a) absence of externalities from action-dispersion
(b) effi ciency in use of information
Bounded recall: reallocation of attention towards sources withintermediate transparency
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Conclusions
Attention in large economies with
- complementarity / substitutability in actions
- rich set of payoff interdependencies
- rich information structure
Effi ciency in allocation of attention requires
(a) absence of externalities from action-dispersion
(b) effi ciency in use of information
Bounded recall: reallocation of attention towards sources withintermediate transparency
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Conclusions
Future work
- endogenous sources / social learning(e.g., capital mkts → information aggregation)
- "optimal" recall strategy
- dynamics (optimal stopping)
- fully flexible info. structures (attention-based correlated eq.)
Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions
Thank You!